Download Document

Complex Zeros; Fundamental Theorem of Algebra
Objective:
• SWBAT identify complex zeros of a polynomials by using
Conjugate Root Theorem
• SWBAT find all real and complex zeros by using
Fundamental Theorem of Algebra
Complex Zeros; Fundamental Theorem of Algebra
Complex Numbers
The complex number system includes real and imaginary
numbers.
Standard form of a complex number is: a + bi.
a and b are real numbers.
i is the imaginary unit −1  (𝑖 2 = −1).
Fundamental Theorem of Algebra
Every complex polynomial function of degree 1 or larger
(no negative integers as exponents) has at least one
complex zero.
Complex Zeros; Fundamental Theorem of Algebra
Theorem
Every complex polynomial function of degree n  1 has
exactly n complex zeros, some of which may repeat.
Conjugate Pairs Theorem
If 𝑟 = 𝑎 + 𝑏𝑖 is a zero of a polynomial function whose
coefficients are real numbers, then the complex
conjugate 𝑟 = 𝑎 − 𝑏𝑖 is also a zero of the function.
Examples
1) A polynomial function of degree three has 2 and 3 + i
as it zeros. What is the other zero?
𝑥 =3−𝑖
Complex Zeros; Fundamental Theorem of Algebra
Examples
2) A polynomial function of degree 5 has 4, 2 + 3i, and 5i
as it zeros. What are the other zeros?
𝑥 = 2 − 3𝑖
𝑎𝑛𝑑
𝑥 = −5𝑖
3) A polynomial function of degree 4 has 2 with a zero
multiplicity of 2 and 2 – i as it zeros. What are the
zeros?
𝑎𝑛𝑑 𝑥 = 2 + 𝑖
𝑥 = 2 𝑟𝑒𝑝𝑒𝑎𝑡𝑠 𝑡𝑤𝑖𝑐𝑒
Complex Zeros; Fundamental Theorem of Algebra
Examples
4) A polynomial function of degree 4 has 2 with a zero
multiplicity of 2 and 2 – i as it zeros. What is the function?
𝑥=2
𝑥=2
𝑥 =2−𝑖
𝑥 =2+𝑖
𝑓 𝑥 = (𝑥 − 2)(𝑥 − 2)(𝑥 − (2 − 𝑖))(𝑥 − (2 + 𝑖))
𝑓 𝑥 = (𝑥 2 −4𝑥 + 4)(𝑥 − 2 + 𝑖)(𝑥 − 2 − 𝑖)
(𝑥 2 − 2𝑥 − 𝑖𝑥 − 2𝑥 + 4 + 2𝑖 + 𝑖𝑥 − 2𝑖 − 𝑖 2 )
𝑓 𝑥 = (𝑥 2 −4𝑥 + 4)(𝑥 2 − 4𝑥 + 5)
𝑓 𝑥 = 𝑥 4 − 4𝑥 3 + 5𝑥 2 − 4𝑥 3 + 16𝑥 2 − 20𝑥 + 4𝑥 2 − 16𝑥 + 20
𝑓 𝑥 = 𝑥 4 − 8𝑥 3 + 25𝑥 2 − 36𝑥 + 20
Complex Zeros; Fundamental Theorem of Algebra
Find the remaining complex zeros of the given polynomial
functions
5) 𝑓 𝑥 = 𝑥 3 + 3𝑥 2 + 25𝑥 + 75 𝑧𝑒𝑟𝑜: −5𝑖
𝐴𝑛𝑜𝑡ℎ𝑒𝑟 𝑧𝑒𝑟𝑜 (𝑡ℎ𝑒 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒): 5𝑖
𝑥 = −5𝑖 𝑎𝑛𝑑 𝑥 = 5𝑖
(𝑥 + 5𝑖)(𝑥 − 5𝑖)
𝑥 2 − 5𝑖𝑥 + 5𝑖𝑥 − 25𝑖 2
𝑥 2 − 25(−1)
𝑥 2 + 25
Complex Zeros; Fundamental Theorem of Algebra
𝑓 𝑥 = 𝑥 3 + 3𝑥 2 + 25𝑥 + 75
𝑧𝑒𝑟𝑜: −5𝑖
Long Division
𝑥 +3
x  25 x  3 x  25 x  75
2
3
2
𝑥3
25𝑥
3𝑥 22
3𝑥
+75
+75
0
𝑥 + 3 𝑥 2 + 25
(𝑥 + 3)(𝑥 + 5𝑖)(𝑥 − 5𝑖)
𝑧𝑒𝑟𝑜𝑠: −3, −5𝑖 𝑎𝑛𝑑 5𝑖
Complex Zeros; Fundamental Theorem of Algebra
Find the complex zeros of the given polynomial functions
6) 𝑓 𝑥 = 𝑥 4 − 4𝑥 3 + 9𝑥 2 − 20𝑥 + 20
𝑝: ±1, ±2, ±4, ±5, ±10, ±20
𝑞: ±1
𝑝
1 2 4 5 10 20
: ± ,± ,± ,± ,± ,±
𝑞
1 1 1 1
1
1
Possible solutions: 𝑥 = ±1, ±2, ±4, ±5, ±10, ±20
Try: 𝑥 = −1
Try: 𝑥 = 1
1 1  4 9  20 20
1
6 −14
1 −3
−3 6 −14 6
 1 1  4 9  20 20
1
−1 5 −14
−5 14 −34
34
54
Complex Zeros; Fundamental Theorem of Algebra
𝑓 𝑥 = 𝑥 4 − 4𝑥 3 + 9𝑥 2 − 20𝑥 + 20
Try: 𝑥 = 2
2 1  4 9  20 20
10 −20
2 −4
1 −2 5 −10 0
𝑓 𝑥 = (𝑥 − 2)(𝑥 3 − 2𝑥 2 + 5𝑥 − 10)
𝑓 𝑥 = (𝑥 − 2)(𝑥 2 𝑥 − 2 + 5(𝑥 − 2))
𝑓 𝑥 = (𝑥 − 2)(𝑥 − 2)(𝑥 2 + 5)
Complex Zeros; Fundamental Theorem of Algebra
𝑓 𝑥 = 𝑥 4 − 4𝑥 3 + 9𝑥 2 − 20𝑥 + 20
𝑓 𝑥 = 𝑥 − 2 𝑥 − 2 𝑥2 + 5 = 0
𝑥=2
𝑥2 + 5 = 0
𝑥 2 = −5
𝑧𝑒𝑟𝑜 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 2
𝑥 = ± −5
𝑥−2=0
𝑥−2=0
𝑥 =± 5𝑖
Complex zeros: 2 with multiplicity of 2, 5 𝑖, 𝑎𝑛𝑑 − 5 𝑖
𝑓 𝑥 𝑖𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑒𝑑 𝑓𝑜𝑟𝑚
𝑓 𝑥 = (𝑥 − 2)2 𝑥 − 5 𝑖 𝑥 + 5 𝑖